FIND THE DISTANCE  BETWEEN TWO POINTS USING FORMULA

Let (x₁, y₁) and (x₂, y₂) be the two points as shown below. 

Then, the formula for the distance between the two points is 

√[(x₂ - x₁)² + (y₂ - y₁)²]

Practice Questions

Question 1 :

Find the distance between the following pairs of points.

(1, 2) and (4, 3)

Answer :

Distance between two points  =  √(x₂ - x₁)² + (y₂ - y₁)²

  =  √(4 - 1)² + (2 - 3)²

  =  √3² + (-1)²

  =  √(9 + 1)

  =  √10

Question 2 :

Find the distance between the following pairs of points.

(3, 4) and (– 7, 2)

Answer :

Distance between two points  =  √(x₂ - x₁)² + (y₂ - y₁)²

  =  √(-7 - 3)² + (2 - 4)²

  =  √(-10)² + (-2)²

  =  √(100 + 4)

  =  √104

=  2 √26

Question 3 :

Find the distance between the following pairs of points.

(a, b) and (c, b)

Answer :

Distance between two points  =  √(x₂ - x₁)² + (y₂ - y₁)²

  =  √(c - a)² + (b - b)²

  =  √(c - a)² + 0²

  =  (c - a)

Question 4 :

Find the distance between the following pairs of points.

(3, -9) and (-2, 3)

Answer :

Distance between two points  =  √(x₂ - x₁)² + (y₂ - y₁)²

  =  √(-2 - 3)² + (3 - (-9))²

  =  √(-5)² + (3+9)²

  =  √25 + 144

  =  √169

  =  13

Question 5 :

Determine whether the given set of points in each case are collinear or not.

(7, –2),(5, 1),(3, 4)

Answer :

Let the given points be A (7, –2) B (5, 1) and C (3, 4)

Distance between the points A and B :

  =  √(5 - 7)² + (1 - (-2))²

  =  √(-2)² + (1+2)²

  =  √4 + 9

  =  √13

Distance between the points B and C :

  =  √(3 - 5)² + (4 - 1)²

  =  √(-2)² + (3)²

  =  √4 + 9

  =  √13

Distance between the points C and A :

  =  √(3 - 7)² + (4 - (-2))²

  =  √(-4)² + (6)²

  =  √16 + 36

  =  √52

=  2√13

√13 + √13  =  2√13

So, the given points are collinear.

Question 6 :

Determine whether the given set of points in each case are collinear or not.

(a, –2), (a, 3), (a, 0)

Answer :

Let the given points be A (a, –2) B (a, 3) and C (a, 0)

Distance between the points A and B :

  =  √(a - a)² + (3 - (-2))²

  =  √0 + (5)²

  =  √0 + 25

  =  √25

=  5

Distance between the points B and C :

  =  √(a - a)² + (0 - 3)²

  =  √(0)² + (3)²

  =  √9

  =  3

Distance between the points C and A :

  =  √(a - a)² + (0 - (-2))²

  =  √(0)² + (2)²

  =  √4

  =  2

BC + CA  =  AB

3 + 2  =  5

5  =  5

So, the given points are collinear.

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